MA5 Geometry & Trigonometry

The ACT gives you 45 mathematics questions and 50 minutes, four answer choices each, and — this is the part students underestimate — no formula sheet. Geometry is where that bites, because geometry is the topic with the most formulas. The good news is that the list you actually need is short, and most of it you already half-know.

Worth knowing cold, because nobody will hand them to you: triangle area ½bh · trapezoid area ½(b₁ + b₂)h · circle C = 2πr and A = πr² · interior angles of an n-gon (n − 2)180° · exterior angles of a convex polygon add to 360° · a² + b² = c² · distance √((x₂ − x₁)² + (y₂ − y₁)²) · midpoint ((x₁ + x₂)/2, (y₁ + y₂)/2) · slope (y₂ − y₁)/(x₂ − x₁) · circle in the plane (x − h)² + (y − k)² = r² · cylinder V = πr²h · cone V = ⅓πr²h · sphere V = ⁴⁄₃πr³ and SA = 4πr² · pyramid V = ⅓(base area)(height) · SOHCAHTOA.

No picture? Draw one.

Many ACT geometry questions describe a figure in words: a ladder against a wall, a circle inscribed in a square, a transversal crossing two parallel lines. Read the sentence once, then draw it before you do any arithmetic. Label every number as you place it, and mark the right angles. Half the errors in this topic are not errors of method — they are a length written on the wrong side of the triangle.

Two habits pay for themselves:

Check what the formula wants. Area of a circle wants a radius. If the question gave you a diameter, halve it first — and if you square the diameter by mistake, your answer comes out four times too big, which is why that distractor is always sitting there waiting for you.
Check what the question asked for. Solving for x is usually the second-to-last step. If the question asks for an angle measure, you still have to substitute x back in.

The two triangles worth memorising

Two right triangles turn up constantly, and knowing them turns a calculator problem into a five-second one.

45°-45°-90°: sides in the ratio 1 : 1 : √2. Two equal legs; the hypotenuse is a leg times √2. Going backwards, a leg is the hypotenuse divided by √2.
30°-60°-90°: sides in the ratio 1 : √3 : 2. The side opposite the 30° angle is the short one — exactly half the hypotenuse — and the side opposite the 60° angle is that short side times √3.

The same is true of the common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and every multiple of them (6-8-10, 9-12-15, 10-24-26). If two sides of a right triangle look like a triple, check whether they are — you will save the squaring and the square root.

SOHCAHTOA

Right-triangle trigonometry is three ratios, all measured from a chosen angle:

SOH: sine = opposite ÷ hypotenuse
CAH: cosine = adjacent ÷ hypotenuse
TOA: tangent = opposite ÷ adjacent

The hypotenuse never changes — it is always across from the right angle — but which leg is "opposite" and which is "adjacent" depends on which angle you are working from. That is what makes sin A and cos B the same number in a right triangle, and it is where most trig mistakes start.

Choose the ratio by looking at what you have and what you want. Two legs? Tangent. A leg and the hypotenuse? Sine or cosine. If the unknown ends up in the denominator, as in sin 28° = 9/h, multiply and divide to free it: h = 9 ÷ sin 28°.

Angles of elevation and depression are just this with a story attached. Both are measured from the horizontal, and the angle of depression from a cliff top down to a boat equals the angle of elevation from the boat back up to the cliff top, because those are alternate interior angles across two horizontal lines.

Circles, arcs and sectors

An arc or a sector is a fraction of a whole circle, and the fraction is the central angle over 360°. So an arc is (θ/360)(2πr) and a sector is (θ/360)(πr²). Keep the two straight by asking whether the answer should be a length or an area. Two more facts earn their keep: an inscribed angle is half the central angle standing on the same arc, and a tangent meets the radius at the point of contact at 90° — which turns almost every tangent question into a Pythagorean one.

The trap: "the other number in the problem." A parallelogram has a base of 12, a slanted side of 9, and a height of 7. Its area is 12 × 7 = 84, not 12 × 9 = 108 — area uses the perpendicular height, and the slanted side is longer than that. But do not overcorrect into a rule that is false. In a right triangle the two legs are a base and a height, so there a side really is the height. And 2(base + height) genuinely is the perimeter of a rectangle — a rectangle is a parallelogram whose sides already meet at right angles, so there the "height" is a side. The question to ask is not "is this a side?" but "is this length perpendicular to the base?"

Coordinate geometry is algebra wearing a hat

Distance is the Pythagorean theorem with the legs measured along the axes. Midpoint is an average. Parallel lines have equal slopes; perpendicular lines have slopes that multiply to −1, which means you flip the fraction and change the sign — both steps. And the circle equation (x − h)² + (y − k)² = r² subtracts the coordinates of the centre, so (x + 3)² means the centre's x-coordinate is −3, not 3. Read the sign off the bracket, then flip it.

On the day

A calculator is permitted on the mathematics section and no question requires one. Answers in terms of π — 36π rather than 113.1 — are common in geometry, and when they appear you should stop before decimalising: the exact form is the answer. Finally, four choices means a wrong answer costs you nothing beyond the point, since there is no penalty for guessing. If a solid or a circle question is eating your clock, put down your best estimate and move on.

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